Sharp well-posedness for a coupled system of mKdV type equations

2020/03/27 by Carvajal, Xavier, Esquivel, Liliana, Santos, Raphael
#35Q35 #35Q53 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.12619

Abstract

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations ∂_tv + ∂_x³v + ∂_x(vw²) =0, v(x,0)=ϕ(x), ∂_tw + α∂_x³w + ∂_x(v²w) =0, w(x,0)=ψ(x), and prove the local well-posedness results for given data in low regularity Sobolev spaces H^s(\textrmI \textrmR)× H^k(\textrmI \textrmR), s,k> -\frac12 and |s-k|≤ 1/2, for α≠ 0,1. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be C³ at the origin, when s2; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) s-2k>1 or k1 or s

Related

• Sharp well-posedness for a coupled system of mKdV type equations
• Local well-posedness for a system of modified KdV equations in modulation spaces
• Global Well-Posedness for a Coupled Modified KdV System
• Sharp Global well-posedness for KdV and modified KdV on \R and \T
• Low regularity bounds for mKdV